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## G = C4.D56order 448 = 26·7

### 1st non-split extension by C4 of D56 acting via D56/D28=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C28 — C4.D56
 Chief series C1 — C7 — C14 — C2×C14 — C2×C28 — C4×C28 — C28⋊4D4 — C4.D56
 Lower central C7 — C2×C14 — C2×C28 — C4.D56
 Upper central C1 — C22 — C42 — C4⋊C8

Generators and relations for C4.D56
G = < a,b,c | a4=b56=1, c2=a, bab-1=a-1, ac=ca, cbc-1=ab-1 >

Subgroups: 708 in 84 conjugacy classes, 33 normal (31 characteristic)
C1, C2, C2, C4, C4, C22, C22, C7, C8, C2×C4, D4, C23, D7, C14, C42, C2×C8, C2×D4, C28, C28, D14, C2×C14, C4⋊C8, C4⋊C8, C41D4, C7⋊C8, C56, D28, C2×C28, C22×D7, C4.D8, C2×C7⋊C8, C4×C28, C2×C56, C2×D28, C2×D28, C28⋊C8, C7×C4⋊C8, C284D4, C4.D56
Quotients: C1, C2, C4, C22, C2×C4, D4, D7, C22⋊C4, D8, SD16, D14, C4.D4, D4⋊C4, C4×D7, D28, C7⋊D4, C4.D8, C56⋊C2, D56, D14⋊C4, D4⋊D7, Q8⋊D7, C14.D8, C2.D56, C28.46D4, C4.D56

Smallest permutation representation of C4.D56
On 224 points
Generators in S224
```(1 88 196 167)(2 168 197 89)(3 90 198 113)(4 114 199 91)(5 92 200 115)(6 116 201 93)(7 94 202 117)(8 118 203 95)(9 96 204 119)(10 120 205 97)(11 98 206 121)(12 122 207 99)(13 100 208 123)(14 124 209 101)(15 102 210 125)(16 126 211 103)(17 104 212 127)(18 128 213 105)(19 106 214 129)(20 130 215 107)(21 108 216 131)(22 132 217 109)(23 110 218 133)(24 134 219 111)(25 112 220 135)(26 136 221 57)(27 58 222 137)(28 138 223 59)(29 60 224 139)(30 140 169 61)(31 62 170 141)(32 142 171 63)(33 64 172 143)(34 144 173 65)(35 66 174 145)(36 146 175 67)(37 68 176 147)(38 148 177 69)(39 70 178 149)(40 150 179 71)(41 72 180 151)(42 152 181 73)(43 74 182 153)(44 154 183 75)(45 76 184 155)(46 156 185 77)(47 78 186 157)(48 158 187 79)(49 80 188 159)(50 160 189 81)(51 82 190 161)(52 162 191 83)(53 84 192 163)(54 164 193 85)(55 86 194 165)(56 166 195 87)
(1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56)(57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112)(113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168)(169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224)
(1 124 88 209 196 101 167 14)(2 13 168 100 197 208 89 123)(3 122 90 207 198 99 113 12)(4 11 114 98 199 206 91 121)(5 120 92 205 200 97 115 10)(6 9 116 96 201 204 93 119)(7 118 94 203 202 95 117 8)(15 166 102 195 210 87 125 56)(16 55 126 86 211 194 103 165)(17 164 104 193 212 85 127 54)(18 53 128 84 213 192 105 163)(19 162 106 191 214 83 129 52)(20 51 130 82 215 190 107 161)(21 160 108 189 216 81 131 50)(22 49 132 80 217 188 109 159)(23 158 110 187 218 79 133 48)(24 47 134 78 219 186 111 157)(25 156 112 185 220 77 135 46)(26 45 136 76 221 184 57 155)(27 154 58 183 222 75 137 44)(28 43 138 74 223 182 59 153)(29 152 60 181 224 73 139 42)(30 41 140 72 169 180 61 151)(31 150 62 179 170 71 141 40)(32 39 142 70 171 178 63 149)(33 148 64 177 172 69 143 38)(34 37 144 68 173 176 65 147)(35 146 66 175 174 67 145 36)```

`G:=sub<Sym(224)| (1,88,196,167)(2,168,197,89)(3,90,198,113)(4,114,199,91)(5,92,200,115)(6,116,201,93)(7,94,202,117)(8,118,203,95)(9,96,204,119)(10,120,205,97)(11,98,206,121)(12,122,207,99)(13,100,208,123)(14,124,209,101)(15,102,210,125)(16,126,211,103)(17,104,212,127)(18,128,213,105)(19,106,214,129)(20,130,215,107)(21,108,216,131)(22,132,217,109)(23,110,218,133)(24,134,219,111)(25,112,220,135)(26,136,221,57)(27,58,222,137)(28,138,223,59)(29,60,224,139)(30,140,169,61)(31,62,170,141)(32,142,171,63)(33,64,172,143)(34,144,173,65)(35,66,174,145)(36,146,175,67)(37,68,176,147)(38,148,177,69)(39,70,178,149)(40,150,179,71)(41,72,180,151)(42,152,181,73)(43,74,182,153)(44,154,183,75)(45,76,184,155)(46,156,185,77)(47,78,186,157)(48,158,187,79)(49,80,188,159)(50,160,189,81)(51,82,190,161)(52,162,191,83)(53,84,192,163)(54,164,193,85)(55,86,194,165)(56,166,195,87), (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56)(57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112)(113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168)(169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224), (1,124,88,209,196,101,167,14)(2,13,168,100,197,208,89,123)(3,122,90,207,198,99,113,12)(4,11,114,98,199,206,91,121)(5,120,92,205,200,97,115,10)(6,9,116,96,201,204,93,119)(7,118,94,203,202,95,117,8)(15,166,102,195,210,87,125,56)(16,55,126,86,211,194,103,165)(17,164,104,193,212,85,127,54)(18,53,128,84,213,192,105,163)(19,162,106,191,214,83,129,52)(20,51,130,82,215,190,107,161)(21,160,108,189,216,81,131,50)(22,49,132,80,217,188,109,159)(23,158,110,187,218,79,133,48)(24,47,134,78,219,186,111,157)(25,156,112,185,220,77,135,46)(26,45,136,76,221,184,57,155)(27,154,58,183,222,75,137,44)(28,43,138,74,223,182,59,153)(29,152,60,181,224,73,139,42)(30,41,140,72,169,180,61,151)(31,150,62,179,170,71,141,40)(32,39,142,70,171,178,63,149)(33,148,64,177,172,69,143,38)(34,37,144,68,173,176,65,147)(35,146,66,175,174,67,145,36)>;`

`G:=Group( (1,88,196,167)(2,168,197,89)(3,90,198,113)(4,114,199,91)(5,92,200,115)(6,116,201,93)(7,94,202,117)(8,118,203,95)(9,96,204,119)(10,120,205,97)(11,98,206,121)(12,122,207,99)(13,100,208,123)(14,124,209,101)(15,102,210,125)(16,126,211,103)(17,104,212,127)(18,128,213,105)(19,106,214,129)(20,130,215,107)(21,108,216,131)(22,132,217,109)(23,110,218,133)(24,134,219,111)(25,112,220,135)(26,136,221,57)(27,58,222,137)(28,138,223,59)(29,60,224,139)(30,140,169,61)(31,62,170,141)(32,142,171,63)(33,64,172,143)(34,144,173,65)(35,66,174,145)(36,146,175,67)(37,68,176,147)(38,148,177,69)(39,70,178,149)(40,150,179,71)(41,72,180,151)(42,152,181,73)(43,74,182,153)(44,154,183,75)(45,76,184,155)(46,156,185,77)(47,78,186,157)(48,158,187,79)(49,80,188,159)(50,160,189,81)(51,82,190,161)(52,162,191,83)(53,84,192,163)(54,164,193,85)(55,86,194,165)(56,166,195,87), (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56)(57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112)(113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168)(169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224), (1,124,88,209,196,101,167,14)(2,13,168,100,197,208,89,123)(3,122,90,207,198,99,113,12)(4,11,114,98,199,206,91,121)(5,120,92,205,200,97,115,10)(6,9,116,96,201,204,93,119)(7,118,94,203,202,95,117,8)(15,166,102,195,210,87,125,56)(16,55,126,86,211,194,103,165)(17,164,104,193,212,85,127,54)(18,53,128,84,213,192,105,163)(19,162,106,191,214,83,129,52)(20,51,130,82,215,190,107,161)(21,160,108,189,216,81,131,50)(22,49,132,80,217,188,109,159)(23,158,110,187,218,79,133,48)(24,47,134,78,219,186,111,157)(25,156,112,185,220,77,135,46)(26,45,136,76,221,184,57,155)(27,154,58,183,222,75,137,44)(28,43,138,74,223,182,59,153)(29,152,60,181,224,73,139,42)(30,41,140,72,169,180,61,151)(31,150,62,179,170,71,141,40)(32,39,142,70,171,178,63,149)(33,148,64,177,172,69,143,38)(34,37,144,68,173,176,65,147)(35,146,66,175,174,67,145,36) );`

`G=PermutationGroup([[(1,88,196,167),(2,168,197,89),(3,90,198,113),(4,114,199,91),(5,92,200,115),(6,116,201,93),(7,94,202,117),(8,118,203,95),(9,96,204,119),(10,120,205,97),(11,98,206,121),(12,122,207,99),(13,100,208,123),(14,124,209,101),(15,102,210,125),(16,126,211,103),(17,104,212,127),(18,128,213,105),(19,106,214,129),(20,130,215,107),(21,108,216,131),(22,132,217,109),(23,110,218,133),(24,134,219,111),(25,112,220,135),(26,136,221,57),(27,58,222,137),(28,138,223,59),(29,60,224,139),(30,140,169,61),(31,62,170,141),(32,142,171,63),(33,64,172,143),(34,144,173,65),(35,66,174,145),(36,146,175,67),(37,68,176,147),(38,148,177,69),(39,70,178,149),(40,150,179,71),(41,72,180,151),(42,152,181,73),(43,74,182,153),(44,154,183,75),(45,76,184,155),(46,156,185,77),(47,78,186,157),(48,158,187,79),(49,80,188,159),(50,160,189,81),(51,82,190,161),(52,162,191,83),(53,84,192,163),(54,164,193,85),(55,86,194,165),(56,166,195,87)], [(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56),(57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112),(113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168),(169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224)], [(1,124,88,209,196,101,167,14),(2,13,168,100,197,208,89,123),(3,122,90,207,198,99,113,12),(4,11,114,98,199,206,91,121),(5,120,92,205,200,97,115,10),(6,9,116,96,201,204,93,119),(7,118,94,203,202,95,117,8),(15,166,102,195,210,87,125,56),(16,55,126,86,211,194,103,165),(17,164,104,193,212,85,127,54),(18,53,128,84,213,192,105,163),(19,162,106,191,214,83,129,52),(20,51,130,82,215,190,107,161),(21,160,108,189,216,81,131,50),(22,49,132,80,217,188,109,159),(23,158,110,187,218,79,133,48),(24,47,134,78,219,186,111,157),(25,156,112,185,220,77,135,46),(26,45,136,76,221,184,57,155),(27,154,58,183,222,75,137,44),(28,43,138,74,223,182,59,153),(29,152,60,181,224,73,139,42),(30,41,140,72,169,180,61,151),(31,150,62,179,170,71,141,40),(32,39,142,70,171,178,63,149),(33,148,64,177,172,69,143,38),(34,37,144,68,173,176,65,147),(35,146,66,175,174,67,145,36)]])`

79 conjugacy classes

 class 1 2A 2B 2C 2D 2E 4A 4B 4C 4D 4E 7A 7B 7C 8A 8B 8C 8D 8E 8F 8G 8H 14A ··· 14I 28A ··· 28L 28M ··· 28X 56A ··· 56X order 1 2 2 2 2 2 4 4 4 4 4 7 7 7 8 8 8 8 8 8 8 8 14 ··· 14 28 ··· 28 28 ··· 28 56 ··· 56 size 1 1 1 1 56 56 2 2 2 2 4 2 2 2 4 4 4 4 28 28 28 28 2 ··· 2 2 ··· 2 4 ··· 4 4 ··· 4

79 irreducible representations

 dim 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 4 4 4 4 type + + + + + + + + + + + + + + image C1 C2 C2 C2 C4 D4 D7 D8 SD16 D14 C4×D7 D28 C7⋊D4 C56⋊C2 D56 C4.D4 D4⋊D7 Q8⋊D7 C28.46D4 kernel C4.D56 C28⋊C8 C7×C4⋊C8 C28⋊4D4 C2×D28 C2×C28 C4⋊C8 C28 C28 C42 C2×C4 C2×C4 C2×C4 C4 C4 C14 C4 C4 C2 # reps 1 1 1 1 4 2 3 4 4 3 6 6 6 12 12 1 3 3 6

Matrix representation of C4.D56 in GL4(𝔽113) generated by

 112 0 0 0 0 112 0 0 0 0 76 77 0 0 82 37
,
 102 95 0 0 18 9 0 0 0 0 71 97 0 0 75 42
,
 70 105 0 0 90 43 0 0 0 0 42 16 0 0 64 97
`G:=sub<GL(4,GF(113))| [112,0,0,0,0,112,0,0,0,0,76,82,0,0,77,37],[102,18,0,0,95,9,0,0,0,0,71,75,0,0,97,42],[70,90,0,0,105,43,0,0,0,0,42,64,0,0,16,97] >;`

C4.D56 in GAP, Magma, Sage, TeX

`C_4.D_{56}`
`% in TeX`

`G:=Group("C4.D56");`
`// GroupNames label`

`G:=SmallGroup(448,42);`
`// by ID`

`G=gap.SmallGroup(448,42);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,141,36,422,100,1123,794,136,18822]);`
`// Polycyclic`

`G:=Group<a,b,c|a^4=b^56=1,c^2=a,b*a*b^-1=a^-1,a*c=c*a,c*b*c^-1=a*b^-1>;`
`// generators/relations`

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